The purpose of this thesis is to find upper bounds for the eigenvalues of natural operators acting on functions on a compact Riemannian manifold $(M,g)$ such as the Laplace-Beltrami operator and Laplace-type operators. In the case of the Laplace-Beltrami operator, two aspects are investigated: The first aspect is to study relationships between the intrinsic geometry and eigenvalues of the Laplace-Beltrami operator. In this regard, we obtain upper bounds depending only on the dimension and a conformal invariant called min-conformal volume. Asymptotically, these bounds are consistent with the Weyl law. They improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is powerful and interesting in itself. The second aspect is to study the interplay of the extrinsic geometry and eigenvalues of the Laplace-Beltrami operator acting on compact submanifolds of $R^N$ and of $C P^N$. We investigate an extrinsic invariant called the intersection index studied by Colbois, Dryden and El Soufi. For compact submanifolds of $R^N$, we extend their results and obtain upper bounds which are stable under small perturbation. For compact submanifolds of $C P^N$, we obtain an upper bound depending only on the degree of submanifolds and which is sharp for the first eigenvalue. As a further application of the introduced method, we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with $C^1$ boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the min-conformal volume. A modification of our method also lead to have upper bounds for the eigenvalues of Schrödinger operators in terms of the min-conformal volume and integral quantity of the potential. As another application of our method, we obtain upper bounds for the eigenvalues of the Bakry-Emery Laplace operator depending on conformal invariants and properties of the weighted function.
